Affiliation:
1. School of Mathematics and Statistics and Guizhou University of Finance and Economics , Guiyang , 550025 , P. R. China
2. School of Science and Jiangsu University of Science and Technology , Zhenjiang 212003 , China
Abstract
Abstract
This paper deals with the following weakly coupled nonlinear Schrödinger system
−
Δ
u
1
+
a
1
(
x
)
u
1
=
∣
u
1
∣
2
p
−
2
u
1
+
b
∣
u
1
∣
p
−
2
∣
u
2
∣
p
u
1
,
x
∈
R
N
,
−
Δ
u
2
+
a
2
(
x
)
u
2
=
∣
u
2
∣
2
p
−
2
u
2
+
b
∣
u
2
∣
p
−
2
∣
u
1
∣
p
u
2
,
x
∈
R
N
,
\left\{\begin{array}{ll}-\Delta {u}_{1}+{a}_{1}\left(x){u}_{1}=| {u}_{1}{| }^{2p-2}{u}_{1}+b| {u}_{1}{| }^{p-2}| {u}_{2}{| }^{p}{u}_{1},& x\in {{\mathbb{R}}}^{N},\\ -\Delta {u}_{2}+{a}_{2}\left(x){u}_{2}=| {u}_{2}{| }^{2p-2}{u}_{2}+b| {u}_{2}{| }^{p-2}| {u}_{1}{| }^{p}{u}_{2},& x\in {{\mathbb{R}}}^{N},\end{array}\right.
where
N
≥
1
N\ge 1
,
b
∈
R
b\in {\mathbb{R}}
is a coupling constant,
2
p
∈
(
2
,
2
∗
)
2p\in \left(2,{2}^{\ast })
,
2
∗
=
2
N
/
(
N
−
2
)
{2}^{\ast }=2N\hspace{0.1em}\text{/}\hspace{0.1em}\left(N-2)
if
N
≥
3
N\ge 3
and
+
∞
+\infty
if
N
=
1
,
2
N=1,2
,
a
1
(
x
)
{a}_{1}\left(x)
and
a
2
(
x
)
{a}_{2}\left(x)
are two positive functions. Assuming that
a
i
(
x
)
(
i
=
1
,
2
)
{a}_{i}\left(x)\hspace{0.33em}\left(i=1,2)
satisfies some suitable conditions, by constructing creatively two types of two-dimensional mountain-pass geometries, we obtain a positive synchronized solution for
∣
b
∣
>
0
| b| \gt 0
small and a positive segregated solution for
b
<
0
b\lt 0
, respectively. We also show that when
1
<
p
<
min
{
2
,
2
∗
/
2
}
1\lt p\lt {\rm{\min }}\left\{2,{2}^{\ast }\hspace{0.1em}\text{/}\hspace{0.1em}2\right\}
, the positive solutions are not unique if
b
>
0
b\gt 0
is small. The asymptotic behavior of the solutions when
b
→
0
b\to 0
and
b
→
−
∞
b\to -\infty
is also studied.
Subject
General Mathematics,Statistical and Nonlinear Physics
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献